Download ACOUPLED CHEMO-MECHANICAL MODEL FOR

A COUPLED CHEMO-MECHANICAL MODEL FOR CEMENTED SOILS,
FROM THE MICRO- TO THE MACRO-SCALE
Alessandro Gajo
Università di Trento, Dipartimento di ingegneria civile, ambientale e meccanica, Trento
[email protected]
Francesco Cecinato
Università di Trento, Dipartimento di ingegneria civile, ambientale e meccanica, Trento
[email protected]
Tomasz Hueckel
Duke University, Civil and Environmental Engineering, Durham, NC, USA
[email protected]
Abstract
Chemical effects in the mechanical behavior of geomaterials have received increasing attention in recent years
for a number of applications, ranging from the stability of slopes and coastal structures to geological CO2
sequestration in carbonate rocks. Chemical processes result in either strengthening or weakening effects, the
latter being particularly critical for safety assessment of applications in civil and energy engineering. In this
work, coupling of chemical and mechanical processes in cemented soils and rocks is investigated by means of a
micro-structure inspired model, consisting of an assembly of grains and bonds undergoing dissolution or
deposition of mineral mass, affecting geometrical characteristics of the system and thus determining the
evolution of specific surface area and of bond cross-sectional area at the micro-scale, and of porosity at the
macro-scale, which become key variables linking the micro-scale and macro-scale mechanisms. At the macroscale, a reactive chemo-plasticity model incorporating elasto-plastic coupling for bonded geomaterials is
formulated. The resulting micro- to macro-scale model, schematically applicable to both materials with reactive
grains and bonds and materials with only reacting bonds, is validated against the available experimental
evidence, consisting of a number of different chemo-mechanical experiments on calcarenite. The model is thus
shown to provide a flexible framework for consistent interpretation of experimental loading paths, and can be
readily extended to more complex materials or to incorporate additional chemo-mechanical effects.
1. Introduction
Terzaghi (1950) in his paper on Mechanism of Landslide discusses the chemical weathering affecting
rock of any kind which “weakens intergranular bonds” leading to decrease in cohesion, leading to a
progressive decrease in Factor of Safety, due to chemical weathering. Also in the case of loess, water
from external reservoirs is identified as responsible for removing soluble binders and destroying
intergranular bond(s), with the same macroscopic effect of a decreasing cohesion. In a more modern
day context one needs to add that the dissolution of cementation bonds may be dramatically enhanced
by an increase in acidity of the run-off water, namely from an acid rain (e.g. see Zhao et al. 2011).
Further, the recent interest in geological CO2 sequestration in carbonate rocks brings challenges in
predicting the effectiveness of injections since CO2, upon being injected, dissolves into an in-situ brine
and produces also a weak acid, which in turn reacts with the surrounding carbonate rock (e.g. see
Andre et al. 2010). On the other hand, the technique of acid softening of rocks prior to hydraulic
fracturing in unconventional gas and oil recovery or heat recovery from geothermal reservoirs poses
interesting questions in ensuring the control of the process (Hu & Hueckel 2011).
The aim of this work is to investigate coupling of chemical and mechanical processes involved in
weakening or strengthening of bonded geomaterials by means of a micro-structure inspired model.
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The micro-scale model is calibrated on (but it is not limited to) a regular array of grains and bonds
undergoing dissolution or deposition of mineral mass, which both affect geometrical characteristics of
the system. Simple reactions of dissolution and deposition between pore fluid and solid minerals are
considered, thus complex phenomena involving solid-solid mass exchange are neglected for the sake
of simplicity. Variable geometrical characteristics are used to determine the evolving specific surface
area and the bond resisting cross-sectional area at the micro-scale, and variable porosity at the macroscale, which become key variables linking the micro-scale and macro-scale mechanisms. At the
macro-scale, a reactive chemo-plasticity model (e.g. Loret et al. 2002) is combined with a model for
bonded geomaterials (e.g. Gens & Nova 1993), while kinetic rate equations must be adapted for
dissolution/precipitation accounting for their acidity sensitivity (e.g. Sjoberg 1976).
Schematically, based on the kinetics of dissolution/deposition reaction of different minerals, two main
cases are discussed, namely the case of both reactive grains and reactive bonds (e.g. calcarenite, where
both grains and bonds are made of calcium carbonate) and the case of non-reactive grains and reactive
bonds (e.g. silicic sand with carbonate bonds). While in the former case the timescales of
dissolution/deposition of grains and bonds are comparable, in the latter case the timescale of reaction
of bonds is negligibly small compared to that of the grains. However, the proposed framework could
be easily extended to situations involving the presence of different families of cementation bonds
characterized by different strengths and chemical dissolution properties.
2. Model outline
Two types of information can be obtained from the existing literature: information on the material
strength change and on stiffness change, due to mechanical loading and chemical dissolution - that is
typically provided by macro-scale experiments; and information about the chemical processes of
dissolution and precipitation of the key minerals, which is obtained from micro-scale information.
Consequently, we construct a two-scale model, which includes: (i) a macro-scale chemo-elasto-plastic
model, where both the apparent preconsolidation stress and the apparent isotropic tensile strength
depend on the material microstructural features, and (ii) an integrated model for mass change of
minerals, in the framework of an idealized evolving micro-scale structure of grains and bonds, subject
to removal or addition of mineral mass, localized mechanical failure and chemical healing.
The macro-scale, in the considered case, is described by continuum variables of (elastic and plastic)
strain and stress tensors, continuum free energy, and compressive and tensile strength. Such variables
all depend on a series of macro-scale variables, such as the mass change of the minerals, and on microstructural quantities (as the geometry of bonds and grains). The macroscopic continuum variables
depend through phenomenological functions on micro-structural variables.
The macroscopic constitutive model is developed assuming a yield function along the lines of an
existing approach for bonded geomaterials (e.g. Gens & Nova 1993), in which an isotropic tensile
strength function is included, as an extension of the modified Cam Clay model. An associated flow
rule is adopted for the sake of simplicity. The expression of the yield locus includes two macroscopic
quantities, ptens and pcomp , representing the increase of tensile and compressive strength, compared to
uncemented soil, due to the presence of cementation bonds (Gajo et al. 2015). Coupling with the
micromechanical behavior is introduced by assuming that ptens and pcomp depend on the mean specific
cross section area ab of all mechanically active cementing bonds (corresponding to unbroken bonds,
thus ‘actively’ contributing to the macroscopic strength).
Further, a simplified hardening relationship similar to that of Cam Clay is adopted, assuming
negligible elastic compressibility (Gajo et al. 2015). However, the proposed constitutive framework is
not limited to the above outlined simple assumptions.
The macroscopic strain tensor is decomposed into additive elastic and plastic parts. The macroscopic
elastic behavior of the material is described in the framework of hyperelasticity and elasto-plastic
coupling (Hueckel 1976), and can thus be deduced upon defining a suitable elastic free energy density
function. To account for the presence of both mechanically and chemically interacting bonds, the
standard linear elasticity form of the free energy function of the uncemented solid skeleton (i.e. the
A. Gajo, F. Cecinato & T. Hueckel
Incontro Annuale dei Ricercatori di Geotecnica 2015- IARG 2015
Cagliari, 24-26 giugno 2015
unbonded soil)  g   g  ε e  is modified, based on phenomenological interpretations, into a more
general function:
    ε e , ε p , mb  .
A macroscopic variable mb is the time-integrated mass change of all mechanically active cementing
bonds per unit volume, from a reference configuration mb 0 (  mb  0 implies an increase of bond
mass with respect to the reference configuration).
From the chemical point of view, for the case of only reacting bonds, during cement dissolution a
fraction of cement bonds, which in a general situation carries a fraction of the overall stress,
disappears, causing the reduction of ab . This leads to a stress increase in the remaining fractions of the
bonds, thus inducing strains at constant stress level. As a result, cement dissolution induces a decrease
of soil stiffness with associated strains (if dissolution occurs at constant stress) or a decrease of soil
stiffness with an associated decrease of applied stress (if dissolution occurs at constant strain). On the
other hand, cement deposition induces an increase of material stiffness, at constant stress and strain.
This behavior is taken into account with a step-wise form of the elastic energy (Gajo et al. 2015).
In the case of both reactive solid grains and bonds, the additional consideration must be made that if
the number of cementing bonds is negligible, grain dissolution implies some macroscopic volumetric
strain. In fact, in the limit of null active cementing bonds, grain dissolution is associated to volume
compression (due to the decrease of grain volume). The volumetric effect of grain dissolution in
uncemented soil is completely analogous to the volumetric effect of a decrease of temperature in the
grain material, in contrast with the effect of deposition, which is assumed to produce no volumetric
strain.
From the macroscopic point of view, the net mass change m within the REV due to dissolution and/or
deposition is controlled by chemical kinetics. The rate at which the minerals aggregate or dissociate is
proportional to the specific reactive surface area ar per unit volume of the reacting materials. The
macroscopic quantity ar (with units m2/m3) corresponds to the area per unit volume of all interfaces
of the porous skeleton that are exposed to chemical interaction, i.e. the areas that are actually in
contact with the pore fluid.
Further, a chemo-mechanical rate equation is defined via an empirical function, expressing the rate of
change of mechanically active bonds per unit volume, N ba , including the two mechanisms of
destructuration and cement deposition,
N ba  f  mb , ε p  . As a result, in general cement
dissolution/deposition mb affects soil’s mechanical strength in two ways, namely (i) through the
restoration of previously broken bonds and (ii) through the reduction/increase of the bond cross
sections. In fact, in general cement dissolution/deposition is expected to affect both porosity and the
thickness of bonds.
The constitutive concepts described above are based on two key macroscopic quantities, the cross
section of active bonds ab and the reactive surface area ar , which must be defined in terms of
microscopic variables. To relate the evolution of microscopic variables with the macroscopic chemomechanical description of the material, reference is made to a simplified microscopic geometry. As an
example, in Figure 1a, a 2D schematic of the considered geometry at the microscopic scale inspired
from microscope photographs of thin sections of calcareous rocks is shown, in the general case where
grains might not be in direct contact, but they are linked by cementation bonds that are assumed to be
isotropically distributed. In a large porosity configuration such as that illustrated in Figure 1, the
geometry of bonds can be approximated with that of a cylinder, and the geometry of grains can be
represented by that of a sphere. In Figure 1b, a 2D schematic is shown of the case where, regardless of
the size of bonds, grains remain in direct contact, resulting in a simplification of the bonds’ geometry.
A. Gajo, F. Cecinato & T. Hueckel
( 1)
Incontro Annuale dei Ricercatori di Geotecnica 2015- IARG 2015
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a)
b)
Figure 1. Schematic of the considered simplified geometry at large porosity, (a) in a general case where the
spherical grains are not in direct contact, but they are linked by cylindrical bonds, and (b) in the case where the
spherical grains are in direct contact, and bonded by cylindrical bonds.
The key cross-scale functions that are developed by considering the material microstructure are the
expressions of ar  f Rb , Rg , Lb , d , vb , v g , ε and ab  f Rb , ε p , ar , where Rb , Rg are the mean




bond and grain radii, Lb the mean bond length, d the mean distance between grain edges, vb , vg the
bond and grain specific volume and ε, ε p the total and plastic strain tensor. The interested reader is
referred to Gajo et al. (2015) for further details.
3. Model validation
The above described model was numerically integrated through a fully implicit, backward Euler
integration scheme, showing adequate convergence.
The model capabilities to reproduce the experimental chemo-mechanical behavior of bonded
geomaterials were tested by simulating a series of loading paths along the lines of experimental
datapoints reported in the literature, after having deduced the relevant parameters as much as possible
from published data.
As a first example, the model is tested by simulating unconfined compression tests on saturated
calcarenite, by first reproducing an acid weathering phase and subsequently a mechanical loading
phase, and comparing the simulations with the experimental data of Ciantia et al. (2014). In this
context, a variable expressing the ‘degree of dissolution’ can be introduced as   M dis M 0 , where
M dis represents the dissolved mass and M 0 the initial mass of reacting solids, so that  =0 for the
unweathered material and  =1 at complete dissolution of the reacting solids.
As discussed in section 2, the equivalent elastic modulus of the bonded material E depends on both
the elastic properties of calcite cement and of the unbonded granular material, and is calculated upon
taking the second derivative of the free elastic energy density with respect to elastic strain. Thus, we
obtain an expression of the form
E  E g 1  ab   Eb ab
where E g and Eb the elastic moduli of uncemented solid skeleton and of completely cemented soil
(which is approximated with that of pure calcite) respectively, and parameter  =1.8. It should be
noted that eqn. (2) is also consistent with mixture theory-based rules to obtain the equivalent
parameters of a composite mass, typically employed in soil improvement calculations (e.g. see Ou et
al. 1996). Moreover, eqn. (2) includes elasto-plastic coupling, because the cross section of active
bonds ab depends on plastic strains, which may induce damage of elastic stiffness.
In Figure 2a, simulations of stress-strain curves during unconfined compression tests are plotted at
different values of  , together with analogous experimental data. To adequately reproduce
experimental conditions, the simulation consisted of two steps, namely of a first step reproducing
exposure to an acid solution up to the target value of  , and of a second step simulating the
unconfined compression test in the weathered material. The reproduction of experimental data is
especially accurate until the ultimate strength is attained. After the peak, simulations exhibit a smaller
A. Gajo, F. Cecinato & T. Hueckel
( 2)
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amount of softening than experiments, overall showing a less markedly fragile behavior. Since the
post-peak response is typically deeply affected by shear or compaction banding which leads the
conventional behavior to be much different from that of a homogeneously deforming sample,
additional ad-hoc constitutive assumptions were not introduced for more accurate reproduction of this
phenomenon.
Further, oedometer tests on calcarenite samples previously subjected to different degrees of
weathering (Ciantia et al. 2014) were reproduced. In Figure 2b, satisfactory oedometer test simulations
are shown in terms of vertical strain versus vertical stress, at different degrees of dissolution.
Experimental data from Ciantia et al. (2014) are also reported in the graph for comparison.
Finally, in Figure 3 the stress path of Castellanza and Nova (2004)’s oedometer test on calcarenite in
the  p , q  plane is simulated, and fittingly compared to experimental data. The path A-B represents
the mechanical loading phase. Upon reaching point B, the vertical load is kept constant and the acid
weathering phase starts, gradually causing an increase of radial stress, revealing as a linear decrease of
deviatoric stress with increasing mean stress (path B-C).
In all of the above described examples, satisfactory agreement between the model prediction and the
experimental data is obtained, after adequate parameter calibration (Gajo et al. 2015), suggesting that
the presented model provides a powerful and flexible framework for adequately reproducing different
experimental chemo-mechanical loading paths.
a)
b)
Figure 2. Comparison of simulations and experimental data in terms of (a) deviatoric (axial) stress versus axial
strain in unconfined compression tests on calcarenite, (b) axial strain versus vertical stress in oedometer tests on
calcarenite, carried out after weathering up to different values of degree of dissolution.
Figure 3. Comparison of simulations and experimental data in terms of deviatoric stress versus mean effective
stress in an oedometer test on calcarenite. Path A-B represents the loading phase in unweathered conditions,
path B-C represents the acid weathering phase at constant applied load.
A. Gajo, F. Cecinato & T. Hueckel
Incontro Annuale dei Ricercatori di Geotecnica 2015- IARG 2015
Cagliari, 24-26 giugno 2015
4. Conclusions
In this work an innovative, general two-scale modelling framework able to account for the key aspects
of the coupled chemo-mechanical behavior of bonded granular materials is presented, and validated
against experimental evidence. The proposed framework includes an integrated model for mass
change of minerals, in the frame of an idealized evolving micro-scale structure of grains and bonds,
subject to removal or addition of mineral mass, localized mechanical failure and chemical healing, and
a macro-scale chemo-elasto-plastic model, where both the yield stress under isotropic compression
and the isotropic tensile strength depend on the material microstructural features, incorporating elastoplastic coupling to describe the degradation of elastic properties due to mechanically induced
debonding.
For the sake of simplicity, the distribution of bonds and their cross sections are assumed isotropic and
the model presentation is based on simple linear isotropic elasticity and Cam Clay yield function. The
proposed constitutive framework is not however limited to these simple assumptions and could be
straightforwardly applied to more complex elasticity models and yield functions. Moreover, the
proposed framework could be easily extended to situations involving the presence of different families
of cementation bonds characterized by different strengths and chemical dissolution properties, such as
those experimentally observed by Ciantia et al. (2014) on calcarenite.
It can be concluded that the presented model provides a powerful and flexible framework for
adequately reproducing different experimental chemo-mechanical loading paths. It is worth remarking
that the model is to be intended as an open framework rather than being specialized on a particular
material type. Different additional effects, such as mechanical anisotropy, could easily be incorporated
depending on modeling needs.
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A. Gajo, F. Cecinato & T. Hueckel